# Tall Array Computation

Applying familiar array operations to data larger than memory, with execution deferred and performed in chunks.

**Rests on:** Tensors and Shapes

*MLOps and Platform · MLOps*


## Why learn this

This unlocks the ability to work with datasets that are larger than your computer's memory — a common situation in real machine learning. With tall arrays, you can use the same operations you already know (like computing averages or fitting a model) on data that doesn't fit in RAM. This is a key skill for MLOps because production data is often huge, and you need to handle it without buying a bigger machine or rewriting your code. It leads directly to topics like Model Deployment (moving models to production) and Model Monitoring (watching how they behave on live data), which you've noted as future study areas.


## The intuition

Imagine you have a massive spreadsheet — say, 10 billion rows of customer transactions. Your computer's memory can only hold, maybe, 1 billion rows at a time. A tall array is a way of saying 'this dataset is too big to load all at once, but I still want to ask questions about it.' The key trick is that you don't load it all at once. Instead, you tell the computer what operation you want to do (like 'compute the average purchase amount'), and the computer figures out how to do that operation by reading the data in chunks — a few million rows at a time — and combining the results. Importantly, you write the same code you would write if the data did fit in memory. The tall array machinery handles the chunking and combining for you. It's like having a personal assistant who, instead of asking you to read a 1000-page book in one sitting, reads it chapter by chapter and gives you the summary at the end — you just asked for the summary, not for the whole book at once.


## An analogy

Think of a tall array like a very long conveyor belt of boxes in a warehouse. Each box contains a portion of your data — say, a month's worth of sales. You want to know the total sales for the year. You don't bring all the boxes into your office at once (that would overflow it). Instead, you stand at the end of the belt and, one by one, you open each box, add its contents to your running total, and throw the box away. At the end, you have the yearly total, but you never had the whole dataset in your office. This is exactly how tall arrays work: they process data in chunks, one after another, and combine the partial results. But here's the clever part: the warehouse manager (the tall array engine) doesn't just let you do 'total'. You could also compute the average, the maximum, or even train a model — as long as the operation can be broken down into 'process a chunk, then combine the results'. Where the analogy breaks down: with a conveyor belt, you, the human, have to decide the order and how to combine. With tall arrays, the engine decides that automatically. You just say 'average' and it figures out the rest. Also, the conveyor belt only works if the operation is 'reducible' — you can't sort all the boxes together if the final sorted list is bigger than your office; you'd need a different approach.


## Definition

A tall array is a data structure that represents a dataset too large to fit in memory, letting you run familiar array operations on it by deferring execution until the whole operation is known and then processing the data in chunks, combining the results behind the scenes.


## How it connects

This concept builds directly on MLOps, the parent topic you've already noted in your library. MLOps is about running machine learning in production, and one of the first problems you hit in production is 'my training data is 500 GB and my laptop has 16 GB of RAM'. Tall arrays are the standard answer. It also connects to your notes on Azure ML Studio — that platform provides managed compute and data handling, and under the hood it often uses tall-array-style chunked processing. And it connects to Code Generation for Deployment: once you have a model trained on a tall array, you might compile it into a standalone artifact (like C or ONNX) for an embedded system, and the same chunking discipline can apply there. Your library notes that tall arrays 'extend the same fitting APIs to data that does not fit in memory' — that's exactly the point: you don't learn a new API, you just point your existing code at a tall array and it works.


## The mathematics

This concept does not introduce new mathematics. It relies on arithmetic you already know: adding numbers, dividing them, and comparing them. The method is called 'tall array computation' or, in computer science terms, 'map-reduce'.

The problem it solves is simple. You want to find the average of a list of numbers. To find an average, you must do two things:
1. Add all the numbers together to get a total sum.
2. Count how many numbers there are.
3. Divide the total sum by the count.

The difficulty arises when the list is too long to fit in your computer's memory at once. If you try to load 10,000,000 salaries into memory, your computer will crash. You cannot add them all at once because you cannot see them all at once.

The solution is to split the work into small pieces that fit in memory, process each piece separately, and then combine the results.

Here is how it works with a concrete example. Suppose you want the average salary of 10,000 employees. Your computer can only hold 2,000 salaries in memory at a time.

Step 1: Split the data into chunks.
You divide the 10,000 employees into 5 groups (chunks) of 2,000 employees each.
Chunk 1 contains employees 1 to 2,000.
Chunk 2 contains employees 2,001 to 4,000.
Chunk 3 contains employees 4,001 to 6,000.
Chunk 4 contains employees 6,001 to 8,000.
Chunk 5 contains employees 8,001 to 10,000.

Step 2: Process each chunk independently.
You load Chunk 1 into memory. You add the 2,000 salaries together. Let us say this sum is S1. You also count that there are C1 = 2,000 employees in this chunk. You save S1 and C1. You then discard Chunk 1 from memory.
You load Chunk 2 into memory. You add the 2,000 salaries together to get S2. You count C2 = 2,000. You save S2 and C2. You discard Chunk 2.
You repeat this for Chunks 3, 4, and 5, obtaining sums S3, S4, S5 and counts C3, C4, C5 (all equal to 2,000).

Step 3: Combine the partial results.
Now you have five small pieces of information instead of ten thousand large ones. You do not need the individual salaries anymore. You only need the sums and the counts.
To get the total sum for all employees, you add the partial sums:
Total Sum = S1 + S2 + S3 + S4 + S5.
To get the total count of employees, you add the partial counts:
Total Count = C1 + C2 + C3 + C4 + C5.

Step 4: Compute the final answer.
Average = Total Sum / Total Count.

Why does this work? It works because addition is 'associative'. This is a fancy word for a simple property: it does not matter which numbers you add first. The result is the same whether you add them all at once or in groups.
Mathematically, (a + b) + c = a + (b + c).
Because of this property, the sum of the parts equals the part of the sum.
Sum(Chunk 1) + Sum(Chunk 2) + ... = Sum(All Chunks).

This logic applies to any operation that has this 'associative' property. Examples include:
- Sum: (a+b)+c = a+(b+c). Works.
- Count: (count(a)+count(b))+count(c) = count(all). Works.
- Maximum: max(max(a,b), c) = max(a, max(b,c)). Works.

Operations that do NOT work this way include:
- Average: You cannot average two averages to get the overall average unless the groups are the same size. If Group A has an average of 10 (size 1) and Group B has an average of 20 (size 100), the overall average is not 15. It is much closer to 20. You must keep the sums and counts, as shown above.
- Median: The median of the whole list cannot be found from the medians of the chunks. You would need all the original numbers again.

The tall array engine automates this process. It reads a chunk, computes the partial sum and count, stores those two small numbers, and repeats until all chunks are done. Then it performs one final division.

For any operation f that is associative (meaning combining results in any order gives the same answer), the tall array computation follows this pattern:

final_result = combine( partial_f(chunk_1), partial_f(chunk_2), ..., partial_f(chunk_n) )

Where:
- chunk_i is a small piece of the data that fits in memory.
- partial_f(chunk_i) is the result of applying your operation to just that piece (e.g., the sum of that piece).
- combine is the step where you merge all the partial results together (e.g., adding the sums).
- n is the number of chunks.

The key insight is that you trade memory for time. You do more work (processing chunks separately and then combining) to avoid needing huge amounts of memory.


## A worked example

You have a dataset of daily sales figures for a retail chain. The data is stored as a tall array, meaning it is too large to load all at once.

The dataset has 1,000,000 rows. Each row represents the total sales for one store on one day.
Your computer can only hold 10,000 rows in memory at any given time.
You want to compute the average daily sales figure across all 1,000,000 days.

Because you cannot load 1,000,000 numbers at once, you must split the work into chunks that fit in memory.

Step 1: Determine the number of chunks.
Chunk size = 10,000 rows (the maximum your computer can hold).
Total rows = 1,000,000.
Number of chunks = Total rows / Chunk size = 1,000,000 / 10,000 = 100 chunks.

Step 2: Process the first chunk (Chunk 1).
The engine loads rows 1 to 10,000 into memory.
It adds up all the sales figures in this chunk. Let us say this sum is S1.
S1 = 5,000,000.
It counts the number of rows in this chunk. C1 = 10,000.
The engine saves S1 and C1. It then discards the raw data from memory to make room for the next chunk.

Step 3: Process the second chunk (Chunk 2).
The engine loads rows 10,001 to 20,000 into memory.
It adds up the sales figures. Let us say this sum is S2.
S2 = 4,500,000.
It counts the rows. C2 = 10,000.
The engine saves S2 and C2. It discards the raw data.

Step 4: Process the remaining chunks (Chunks 3 through 100).
The engine repeats this process for each of the remaining 98 chunks.
For simplicity, let us assume that every chunk produces a sum of 4,700,000 and a count of 10,000. (In reality, the sums would vary, but the method is identical).
So, for i = 3 to 100:
Si = 4,700,000.
Ci = 10,000.

Step 5: Combine the partial results.
Now the engine has processed all 100 chunks. It holds only two small lists of numbers: the list of sums and the list of counts.
To find the total sales for the entire year, it adds all the partial sums together.
Total Sum = S1 + S2 + S3 + ... + S100.

Let us calculate this sum step by step:
S1 = 5,000,000
S2 = 4,500,000
S3 to S100 are 98 chunks. Each is 4,700,000.
Sum of S3 to S100 = 98 * 4,700,000.
Calculation: 98 * 4,700,000 = (100 - 2) * 4,700,000 = 470,000,000 - 9,400,000 = 460,600,000.

Total Sum = 5,000,000 + 4,500,000 + 460,600,000.
Total Sum = 9,500,000 + 460,600,000.
Total Sum = 470,100,000.

To find the total number of days, it adds all the partial counts together.
Total Count = C1 + C2 + ... + C100.
Since each Ci is 10,000 and there are 100 chunks:
Total Count = 100 * 10,000 = 1,000,000.

Step 6: Compute the final average.
Average = Total Sum / Total Count.
Average = 470,100,000 / 1,000,000.

To divide by 1,000,000, move the decimal point six places to the left:
470,100,000 becomes 470.1.

Result: The average daily sales figure is 470.1 (say, in ₹).

The tall array engine computed this by reading and processing 100 chunks of 10,000 rows each. At no point did it hold more than 10,000 values in memory. It achieved the correct result using only simple addition and one final division.


## From scratch, in Python

```python
import csv
from typing import Iterator, List

def read_rows_in_chunks(filepath: str, chunk_size: int) -> Iterator[List[float]]:
    """Yield lists of numbers, each list having at most chunk_size elements."""
    chunk = []
    with open(filepath, 'r') as f:
        reader = csv.reader(f)
        for row in reader:
            # assume the first column is a number we care about
            chunk.append(float(row[0]))
            if len(chunk) == chunk_size:
                yield chunk
                chunk = []
    # yield any remaining rows (the last chunk may be smaller)
    if chunk:
        yield chunk

def compute_average(filepath: str, chunk_size: int = 1000) -> float:
    total_sum = 0.0
    total_count = 0
    for chunk in read_rows_in_chunks(filepath, chunk_size):
        # process this chunk: sum and count
        chunk_sum = sum(chunk)
        chunk_count = len(chunk)
        # combine partial results into running totals
        total_sum += chunk_sum
        total_count += chunk_count
    # final combine: divide
    return total_sum / total_count

# Example usage (assuming a file 'sales.csv' exists with one number per line)
# average_sales = compute_average('sales.csv', chunk_size=1000)
# print(average_sales)
```


## The same thing, with a framework

In Python, libraries like Dask or PySpark provide tall-array functionality. With Dask, you write almost the same code as you would with NumPy. For example:

```python
import dask.dataframe as dd

df = dd.read_csv('sales.csv')  # this does NOT load the file; it just records it
mean = df['sales'].mean()
# At this point, nothing has been computed yet.
result = mean.compute()  # this triggers the actual chunked computation
print(result)
```

Underneath, Dask splits the file into chunks, computes the mean of each chunk, and combines those partial means (using the same sum-and-count logic as above). The `compute()` call is what makes it actually run — before that, you're just describing the operation you want. This *deferred evaluation* is exactly the 'execution deferred' part of the definition.


## What the code does

This code shows the core idea of tall arrays in plain Python. First, `read_rows_in_chunks` is a *generator* — a function that yields values one at a time instead of all at once. It opens a CSV file and reads it row by row. It accumulates rows into a `chunk` list. When the chunk reaches `chunk_size` (say, 1000), it yields that list to whoever called it, then resets the chunk to empty. This means the file is never loaded fully into memory — we only ever hold at most `chunk_size` rows at a time. Then, `compute_average` calls this generator in a loop. For each chunk it gets, it computes the sum and the count. These are the 'partial results'. It adds them to running totals (`total_sum` and `total_count`). This is the 'combine' step. After the loop finishes (all chunks processed), it divides the total sum by the total count to get the average. The key point: the computation just needs the partial sums and counts, not all the raw data. The library version (NumPy/PyTorch or MATLAB) does exactly this under the hood, but with more optimisation and better memory management.


## Common misconceptions

**Tall arrays are a different type of data, like a database** — Tall arrays are a special database or a new kind of data format, so I need to learn a whole new data model.

A tall array is just a *view* over existing data (often a file on disk) that presents it as an array-like object. The data itself is still rows and columns; the tall array is just an interface for working with it in chunks without loading it all. In fact: A tall array is a wrapper. You still think of your data as a table or matrix; the tall array handles the chunking and deferred execution for you, using the same APIs as regular in-memory arrays.

**Tall arrays are slow because they process data one row at a time** — Processing in chunks must be much slower than loading everything into memory and doing it in one go.

Tall arrays process data in large chunks (e.g., 1 million rows), not one row at a time. The I/O is sequential and can be parallelized across multiple cores or machines. The overhead is often small compared to the benefit of being able to handle data that would otherwise crash the machine. In fact: Chunked processing can be nearly as fast as in-memory processing when done efficiently, and it's the only option when the data doesn't fit in memory. The bottleneck is usually disk I/O, not the chunking logic.

**Any operation can be done on a tall array** — If I can write `sum` or `mean` with a tall array, I can also write `sort` or a complex rolling window.

Some operations require the whole dataset to be available at once (e.g., a global sort, or computing the median). These cannot be 'reduced' from chunks — you need the original data or a complex multi-pass approach. In fact: Only operations that are associative and reducible work directly on tall arrays. Operations like median, mode, or exact quantiles need special algorithms or are unsupported in the tall-array API.

**Nothing runs until I call compute()** — In Dask/PySpark, if I just write `df.mean()`, the computation starts immediately and I get the answer.

In Dask (and similar frameworks), calling `mean()` builds a *lazy* computation graph — nothing is executed. Only when you call `.compute()` does the actual work happen. In fact: The code is deferred (lazy). You build the entire operation chain first, then trigger execution with `.compute()`. This is a feature — it lets the engine optimize the whole computation before running it.

**Tall arrays store the data in memory but compressed** — The tall array still reads everything into RAM when I create it; it just compresses the data.

Creating a tall array in Dask or MATLAB does *not* load the data. It just stores the file path and the schema. Only when you call compute() does it start reading chunks. In fact: Deferred evaluation means the data stays on disk until you explicitly trigger execution. The tall array object itself is just metadata plus a plan for computation.

**Tall arrays are the same as out-of-core or memory-mapped files** — Tall arrays are just memory-mapped files — I can open a big file on disk and index into it like a normal array.

Memory-mapped files (like `numpy.memmap`) give you random access to a file as if it were an array, but they *do* load parts on demand and don't do deferred optimization across operations. Tall arrays are about *deferred execution and chunked computation*, not just low-level memory mapping. In fact: Memory-mapped files are a building block that tall arrays might use underneath, but tall arrays add the crucial layer of logical (deferred) execution, automatic chunking, and operation fusion.


## In short

A tall array lets you work with data larger than memory using the same code you'd write for in-memory data. Execution is deferred until you ask for the result, then the engine processes the data in chunks, computing partial results and combining them. The key mathematical idea is that *some* operations (like sum and average) can be computed from partial chunks without seeing all data at once — but not all operations (like median). In practice, you write Dask, Spark, or MATLAB code that looks almost ordinary; you just avoid operations that can't be reduced, and you remember to trigger execution with `.compute()`.


## Where this is used

Tall arrays are used in every serious machine learning workflow where the dataset exceeds RAM. Here's how, in concrete terms.

**Large Language Models (LLMs):** LLMs are trained on enormous corpora — trillions of tokens. These are far larger than the memory of any single GPU or even a cluster's RAM. Training pipelines (like those for GPT-style models) use tall-array-like chunking: the training data is stored on disk, and the training loop reads batches of data (e.g., 2 million tokens at a time), computes a gradient update, and moves to the next batch. The model's weights are updated incrementally — no single step ever sees the whole dataset. The 'tall array' here is the tokenized training set, and the 'operation' is the gradient descent update. The developer writes a training loop that calls a data loader, which fetches the next batch — the chunking is implicit but essential. If you tried to load the entire corpus into RAM, the process would crash immediately.

**Small Language Models (on phones/laptops):** When you run a small model on-device (like a keyword spotter or an on-device translation model), you're usually *not* training — you're inferring. But the *training* of those small models (e.g., DistilBERT or TinyLlama) happens on big clusters with tall-array-style data handling. The distinction: tall arrays are a *training-time* concern. For a phone app, you just load the small weights (a few hundred MB) into memory. But the person who trained that model faced the same chunking problem. A developer working with on-device models might use tall arrays to train a custom small model on a large personal dataset (e.g., all their photos) before compressing it for deployment.

**Retrieval-Augmented Generation (RAG):** RAG systems retrieve relevant documents to give the LLM context. The retrieval index often contains millions or billions of embeddings (vectors). Building that index requires processing the entire document corpus — which doesn't fit in memory. The embeddings are computed via a tall-array pipeline: process chunks of documents, compute embeddings, and accumulate them into an index file on disk. The similarity search (e.g., finding the top-k nearest vectors) is also done in chunks — you can't load the whole index into RAM. Libraries like Faiss handle this with chunked processing and memory-mapped files. A developer building a RAG system uses tall arrays (via tools like Dask or Spark) to build the embedding index from a large PDF corpus before the LLM is ever queried.

**Agentic systems:** These are systems that plan, call tools, and act in loops — often over a long period. They typically don't use tall arrays directly because the state they operate on (the current task, the tools available) is small. But the *history* of an agent's actions over weeks could grow large, and if you want to log and analyse that history (e.g., for monitoring or improvement), you'd use tall arrays to analyze the logs without loading months of interaction data into memory. For instance, you might compute the average response time of the agent across 50 million logged actions using a tall array.

**Everyday applied ML (recommendation, vision, forecasting, fraud detection):** These are the classic use cases. A recommendation system for an e-commerce site (like Amazon or Flipkart) has millions of users and products, and the interaction matrix (user × product) is enormous — often terabytes. Training a collaborative filtering model on that matrix is done with tall arrays: chunk the matrix, update the model on each chunk. A fraud detection system processes millions of transactions per day; the training data accumulates and grows beyond RAM quickly. Tall arrays let you retrain your model on all historical transactions without sampling or discarding data. The MATLAB documentation you have (your library source) explicitly says: 'tall arrays extend the same fitting APIs to data that does not fit in memory.' A data scientist in MATLAB writes `mdl = fitlm(tall_array)` and it works, even if the array has a billion rows. In Python, the equivalent is `model.fit` on a Dask or Spark DataFrame — the same API, the underlying chunking is handled automatically.

**What building software with this looks like:** In practice, a developer reaches for Dask (Python), Spark, or the tall-array features in MATLAB. They write code that looks almost identical to in-memory code. The main differences: (1) they often explicitly set the chunk size or let the engine guess; (2) they are mindful of what operations are "reducible" — a sum works, but a sort or a rolling window may not; (3) they call `.compute()` (in Dask) to trigger execution — nothing actually runs until then. What breaks when it's done badly: if you accidentally trigger a `.compute()` on a huge array without a reduction, you'll run out of memory and your machine will freeze or crash. Another classic failure: you try to do an operation that isn't chunkable (like a global sort) and the engine either refuses or silently spills to disk, becoming very slow. The 'scope creep' of an operation — e.g., computing a rolling average over a large window — can force large shuffles of data between chunks, killing performance.


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Read this on the web: https://basavapurushottam.com/learn-ai/tall-array-computation
